The use of articles is one of the hardest things for Chinese speakers to master, since there are so many rules and usages of “the”, and so many exceptions to these rules.
In the example you give, articles are generally not used before names (e.g. Taiwan, Taipei 101, Frank, Jill), but there are exceptions: the Sudan, the Ukraine, the Gambia, and plurals like the Philippines and the Maldives. And Donald Trump is known as “the Donald”.
(The following type of exchange has always irked me: A: I went to Miaoli today. B: What’s a Miaoli? It irks me because a name like Miaoli should not be preceded by an article.)
Now: is MRT a name? In Taipei we take the MRT, while in San Francisco we take BART.
“the” is generally used before things that are definite - meaning both the speaker and the listener know what it is. The indefinite article “a” is often used to introduce an item to the listener; “the” is used thereafter. “I saw a guy on a Harley today. The guy had tattoos on his arms.”
“the” also makes things more specific/concrete, while omitting it makes it more general/abstract: compare the presence/absense of “the” before “Congress” and “society”, or phrases like “over hill and dale”. But “the” can also make things more abstract in some circumstances: “Let us contemplate the hand”; “Before the invention of the wheel…”.
“the” also makes things unique, such as “I’m the doctor at Mucha Health Clinic”, which means that I’m the only doctor there. “I’m a doctor at Mucha Health Clinic” implies that there are other doctors.
Omission of “the” can emphasize that something is an institution that one belongs to: compare “I am going to prison” with “I am going to the prison”. (I’d rather say the latter!) Words like “school”, “church”, “college” and in the UK, “hospital” and “university” are often not preceded by “the”, giving it a different meaning than if preceded by “the”. Further complicating this is “home”, which functions as an adverb: “I am going home”. Compare the meanings of the following:
“I am going home” (i.e. returning to my residence)
“I am going to a home” (i.e. being committed to an [perhaps mental] institution)
“I am going to the home” (i.e. going to a certain institution, perhaps to work there or visit a patient there)
Diseases: we catch meningitis and AIDS, but the flu and the clap.
An entire book can be written on the use of “the”. There is no simple answer on when and when not to use it.
The other posters gave some good advice; I’m just trying to highlight why the Chinese have such a hard time with “the”.